Vortex Lattices in the Superconducting Phases of Doped Topological Insulators and Heterostructures

TL;DR

This paper investigates vortex lattices in doped topological insulators with superconductivity, confirming the existence of Majorana vortex states beyond semi-classical limits and analyzing their behavior across different chemical potentials.

Contribution

It provides a self-consistent analysis of Majorana vortex states in topological insulators with superconductivity, extending understanding beyond semi-classical approximations.

Findings

Majorana vortex states appear when the chemical potential is tuned across the band edge.

The vortex phase transition persists beyond semi-classical limits.

Majorana modes hybridize and tunnel between surfaces depending on chemical potential.

Abstract

Majorana fermions are predicted to play a crucial role in condensed matter realizations of topological quantum computation. These heretofore undiscovered quasiparticles have been predicted to exist at the cores of vortex excitations in topological superconductors and in heterostructures of superconductors and materials with strong spin-orbit coupling. In this work we examine topological insulators with bulk s-wave superconductivity in the presence of a vortex-lattice generated by a perpendicular magnetic field. Using self-consistent Bogoliubov-de Gennes, calculations we confirm that beyond the semi-classical, weak-pairing limit that the Majorana vortex states appear as the chemical potential is tuned from either side of the band edge so long as the density of states is sufficient for superconductivity to form. Further, we demonstrate that the previously predicted vortex phase transition survives beyond the semi-classical limit. At chemical potential values smaller than the critical chemical potential, the vortex lattice modes hybridize within the top and bottom surfaces giving rise to a dispersive low-energy mid-gap band. As the chemical potential is increased, the Majorana states become more localized within a single surface but spread into the bulk toward the opposite surface. Eventually, when the chemical potential is sufficiently high in the bulk bands, the Majorana modes can tunnel between surfaces and eventually a critical point is reached at which modes on opposite surfaces can freely tunnel and annihilate leading to the topological phase transition previously studied in the work of Hosur et al.